Level 1 — RecognitionCalculus III — Sequences & Series

Calculus III — Sequences & Series

⏱ 20 minutes30 marksprintable — key stays hidden on paper

Level 1 Paper — Recognition

Time limit: 20 minutes Total marks: 30

Section A — Multiple Choice (1 mark each)

Select the single best answer.

Q1. The sequence $a_n = \dfrac{3n^2+1}{2n^2+5}$ converges to: (a) $0$ (b) $\dfrac{3}{2}$ (c) $\infty$ (d) $1$

Q2. The geometric series $\displaystyle\sum_{n=0}^{\infty} \left(\tfrac{2}{3}\right)^n$ converges to: (a) $\dfrac{2}{3}$ (b) $\dfrac{3}{2}$ (c) $3$ (d) diverges

Q3. The $p$ -series $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^p}$ converges if and only if: (a) $p>1$ (b) $p\geq 1$ (c) $p<1$ (d) $p>0$

Q4. By the Divergence Test, $\displaystyle\sum_{n=1}^{\infty} \dfrac{n}{n+1}$ : (a) converges to $1$ (b) diverges (c) converges to $0$ (d) test is inconclusive

Q5. For the Ratio Test, the series $\sum a_n$ converges absolutely if $\displaystyle\lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right| = L$ with: (a) $L>1$ (b) $L=1$ (c) $L<1$ (d) $L=0$ only

Q6. The Maclaurin series of $e^x$ is: (a) $\displaystyle\sum_{n=0}^{\infty}\dfrac{x^n}{n!}$ (b) $\displaystyle\sum_{n=0}^{\infty}(-1)^n x^n$ (c) $\displaystyle\sum_{n=0}^{\infty}\dfrac{(-1)^n x^{2n}}{(2n)!}$ (d) $\displaystyle\sum_{n=1}^{\infty}\dfrac{x^n}{n}$

Q7. The radius of convergence of $\displaystyle\sum_{n=0}^{\infty}\dfrac{x^n}{2^n}$ is: (a) $1$ (b) $2$ (c) $\tfrac12$ (d) $\infty$

Q8. The Alternating Series (Leibniz) Test requires the terms $b_n$ to be: (a) increasing and $\to 0$ (b) positive, decreasing, and $\to 0$ (c) bounded only (d) $\to \infty$

Q9. A series that converges but whose absolute-value series diverges is called: (a) absolutely convergent (b) divergent (c) conditionally convergent (d) geometric

Q10. The telescoping series $\displaystyle\sum_{n=1}^{\infty}\left(\dfrac{1}{n}-\dfrac{1}{n+1}\right)$ converges to: (a) $0$ (b) $1$ (c) $2$ (d) diverges

Section B — Matching (1 mark each, Q11 total 5 marks)

Q11. Match each series in Column A to its correct classification in Column B.

Column A	Column B
(i) $\sum \frac{1}{n}$	(P) converges (geometric, $
(ii) $\sum \frac{1}{n^2}$	(Q) diverges ( $p$ -series, $p=1$ )
(iii) $\sum (-1)^{n+1}\frac{1}{n}$	(R) converges ( $p$ -series, $p>1$ )
(iv) $\sum \left(\frac{1}{2}\right)^n$	(S) diverges (divergence test)
(v) $\sum \frac{n}{2n+1}$	(T) conditionally convergent

Section C — True/False WITH justification (2 marks each: 1 for T/F, 1 for reason)

Q12. Every bounded sequence converges. (T/F + justify)

Q13. If $\displaystyle\lim_{n\to\infty} a_n = 0$ , then $\sum a_n$ converges. (T/F + justify)

Q14. The Squeeze Theorem gives $\displaystyle\lim_{n\to\infty}\dfrac{\sin n}{n}=0$ . (T/F + justify)

Q15. Absolute convergence implies convergence. (T/F + justify)

Q16. The Ratio Test with $L=1$ proves the series diverges. (T/F + justify)

Q17. The Maclaurin series of $\cos x$ contains only even powers of $x$ . (T/F + justify)

Q18. A monotonic sequence that is bounded above must converge. (T/F + justify)

Answer keyMark scheme & solutions

Section A (1 mark each)

Q1 — (b) $3/2$ . Divide numerator and denominator by $n^2$ : $\frac{3+1/n^2}{2+5/n^2}\to \frac32$ . (1)

Q2 — (c) $3$ . Geometric with $a=1$ , $r=\tfrac23$ : sum $=\frac{1}{1-2/3}=3$ . (1)

Q3 — (a) $p>1$ . Standard $p$ -series result via integral test. (1)

Q4 — (b) diverges. $\lim \frac{n}{n+1}=1\neq0$ ; divergence test. (1)

Q5 — (c) $L<1$ . Ratio test convergence condition. (1)

Q6 — (a) $\sum x^n/n!$ . Definition of $e^x$ Maclaurin series. (1)

Q7 — (b) $2$ . $|x/2|<1 \Rightarrow |x|<2$ . (1)

Q8 — (b) positive, decreasing, $\to 0$ . Leibniz conditions. (1)

Q9 — (c) conditionally convergent. By definition. (1)

Q10 — (b) $1$ . Partial sum $S_N = 1-\frac{1}{N+1}\to 1$ . (1)

Section B

Q11 — (i)→(Q); (ii)→(R); (iii)→(T); (iv)→(P); (v)→(S). (1 each, 5 total) Reasoning: harmonic ( $p=1$ ) diverges; $p=2>1$ converges; alternating harmonic converges but $\sum 1/n$ diverges ⇒ conditional; $r=\tfrac12$ geometric converges; $\frac{n}{2n+1}\to\tfrac12\neq0$ diverges.

Section C (2 marks each: 1 T/F, 1 reason)

Q12 — False. $a_n=(-1)^n$ is bounded but oscillates and does not converge. (1+1)

Q13 — False. Necessary not sufficient: $\sum 1/n$ has $a_n\to0$ yet diverges. (1+1)

Q14 — True. $-\frac1n \le \frac{\sin n}{n}\le \frac1n$ , both bounds $\to 0$ , so squeeze gives $0$ . (1+1)

Q15 — True. If $\sum|a_n|$ converges then $\sum a_n$ converges (standard theorem). (1+1)

Q16 — False. $L=1$ is inconclusive; the test gives no information. (1+1)

Q17 — True. $\cos x = \sum \frac{(-1)^n x^{2n}}{(2n)!}$ ; all exponents even. (1+1)

Q18 — True. Monotone Convergence Theorem: bounded + monotone ⇒ convergent. (1+1)

[
  {"claim":"Q1 limit of (3n^2+1)/(2n^2+5) is 3/2","code":"n=symbols('n'); result = limit((3*n**2+1)/(2*n**2+5), n, oo) == Rational(3,2)"},
  {"claim":"Q2 geometric sum sum_{n=0}^inf (2/3)^n = 3","code":"n=symbols('n'); result = summation(Rational(2,3)**n,(n,0,oo)) == 3"},
  {"claim":"Q10 telescoping sum sum_{n=1}^inf (1/n-1/(n+1)) = 1","code":"n=symbols('n'); result = summation(1/n-1/(n+1),(n,1,oo)) == 1"},
  {"claim":"Q14 squeeze limit sin(n)/n -> 0","code":"n=symbols('n'); result = limit(sin(n)/n, n, oo) == 0"}
]